Math_301_April_2009

Math_301_April_2009 - t then eliminate v to give the...

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The University of British Columbia Math 301 Final Examination - April 2009 1. [10] Find all values of: (a) ( i 2 ) i (b) ( i i ) 2 (c) arcsinh ( i ) 2. [18] Evaluate the integrals; justify all steps carefully (a) K = p.v Z 0 x 1 / 3 x 2 1 dx (b) J = Z 0 x 1 / 2 log( x ) 1+ x 2 dx 3. [18] (a) Carefully construct a branch of the function g ( z )=[ z (1 z )] 1 / 2 with a branch cut on the interval 0 x 1 such that g ( 1 2 (1 + i )) = 1 2 . (b) Evaluate; justify all steps carefully J = Z 1 0 x 1 2 (1 x ) 1 2 dx. 4. [18] (a) Find the image of the upper half z plane ( y 0) under the conformal mapping s = f ( z ) s = ξ + = f ( z )= z i z + i . (b) Find the image of the interior of the unit circle in the s plane under the conformal mapping w = g ( s ) w = u + iv = g ( s )= s (1 s ) 2 . [Hint: f rst show that the image of the circle is real.] 1
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(c) Simplify the combined mapping, de f ning G ( z ) w = u + iv = g ( f ( z )) = G ( z ) . (1) (d) What is the complex potential F ( z ) of a uniform F ow parallel to the x axis in the upper half z plane? Use the above mapping to f nd the image of this uniform F ow in the w plane de f ned by (1). Represent the images parametrically as u ( t ) ,v
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Unformatted text preview: ( t ) , then eliminate v to give the streamlines in the form u = H ( v ) . Sketch several curves. 5. [18] (a) Find the Fourier transform, justify all steps carefully g ( x ) = e − | x | , − ∞ < x < ∞ . (b) Use a Fourier transform to solve u tt + 2 u t + u = u xx , − ∞ < x < ∞ , < t, u ( x, 0) = e − | x | , u t ( x, 0) = 0 , − ∞ < x < ∞ . Write the solution in the form u ( x, t ) = Z ∞ G ( x, t, ω ) dω. 6. [18] (a) Find the inverse Laplace transform, justify all steps carefully G ( x, t ) = L − 1 ( e − x √ s √ s ) (c) Use a Laplace transform to f nd the solution to the problem: u t = u xx , < x < ∞ , < t, u ( x, 0) = , u (0 , t ) = 1 √ t , t > . You may use the results Z ∞ e − r 2 dr = √ π 2 ; Z ∞ 1 √ u e − au du = r π a . 2...
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This note was uploaded on 01/23/2011 for the course MATH 100,200,30 taught by Professor Dr.alejandrocortas during the Winter '10 term at UBC.

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Math_301_April_2009 - t then eliminate v to give the...

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